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November 30, 2010 2.6 Related Rates Guidelines for Solving Related-Rates Problems 1. Identify all given quantities and quantities to be determined. Make a sketch and label the quantities. 2. Write an equations involving the variables whose rates of change either are given or are to be determined. 3. Using the Chain Rule, implicitly differentiate both sides of the equation with respect to time t. 4. Substitute into the resulting equation all known values for the variables and their rates of change. Then solve for the required rate of change. 1 November 30, 2010 Section 3.1 Extrema on an Interval Definition of Extrema Let f be defined on an interval I containing c. 1. f(c) is the minimum of f on I if f(c) ≤ f(x) for all x in I. 2. f(c) is the maximum of f on I if f(c) ≥ f(x) for all x in I. A min or max of f on I is called the extremum. Extreme Value Theorem If f is continuous on [a, b], then f has both a min and max on [a, b]. Definition of Relative Extrema 1. If there is an open interval containing c on which f(c) is a maximum, then f(c) is called a relative maximum of f, or we say "f has a relative max at 2. If there is an open interval containing c on which f(c) is a minimum, then f(c) is called a relative minimum of f, or we say "f has a relative min at 2 November 30, 2010 Definition of a Critical Number Let f be defined at c. If f '(c) = 0 or if f is not differentiable at c, then c is a critical number of f. Theorem 3.2 If f has a relative min or max at x = c, then c is a critical number of f. Guidelines to find the extrema of a continuous function f on [a, b] 1. Find the critical numbers of f in (a, b), 2. Evaluate f at each critical number in (a, b), 3. Evaluate f at each endpoint of [a, b], 4. The least of these values in the min. The greatest is the max. 3 November 30, 2010 Rolle's Theorem Let f be continuous on [a, b] and differentiable on (a, b). If f(a) = f(b), then there is at least one number c in (a, b) such that f '(c) = 0. f(b) = f(a) a b The Mean Value Theorem If f is continuous on [a, b] and differentiable on (a, b), then there exists a number c in (a, b) such that f(b) − f(a) f '(c) = b−a f(b) f(a) a b 4